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Register a new userTwisted Covariant Noncommutative Self-dual Gravity
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A twisted covariant formulation of noncommutative self-dual gravity is presented. The formulation for constructing twisted noncommutative Yang-Mills theories is used. It is shown that the noncommutative torsion is solved at any order of the $theta$-expansion in terms of the tetrad and some extra fields of the theory. In the process the first order expansion in $theta$ for the Plebanski action is explicitly obtained.
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Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use Seiberg-Witten map. It is shown that the noncommutative torsion constraint is solved by the vanishing of commutative torsion. Finally, the noncommutative corrections to the action are computed up to second order.
We construct a new covariant action for flat self-dual gravity in four spacetime dimensions. The action has just one term, but when expanded around an appropriate background gives rise to a kinetic term and a cubic interaction. Upon imposing the light-cone gauge, the action reproduces the expected chiral interaction of Siegel. The new action is in many ways analogous to the known covariant action for self-dual Yang-Mills theory. There is also a sense in which the new self-dual gravity action exhibits the double copy of self-dual Yang-Mills structure.
A spinless covariant field $phi$ on Minkowski spacetime $M^{d+1}$ obeys the relation $U(a,Lambda)phi(x)U(a,Lambda)^{-1}=phi(Lambda x+a)$ where $(a,Lambda)$ is an element of the Poincare group $Pg$ and $U:(a,Lambda)to U(a,Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincare transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincare transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfeld twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.
We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of dual diffeomorphisms on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS$_3$ algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.
We formulate a noncommutative description of topological half-flat gravity in four dimensions. BRST symmetry of this topological gravity is deformed through a twisting of the usual BRST quantization of noncommutative gauge theories. Finally it is argued that resulting moduli space of instantons is characterized by the solutions of a noncommutative version of the Plebanskis heavenly equation.
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